An Ethnomathematics Exercise in Analyzing and Constructing Ornaments

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Journal of Mathematics & Culture

February 2010 5 (1)

ISSN 1558-5336

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An Ethnomathematics Exercise in Analyzing and Constructing Ornamentsin a Geometry Class

Khayriah Massarwe1, Igor Verner1 and Daoud Bshouty2

1Department of Education in Technology and Science

2Faculty of Mathematics

Technion Israel Institute of Technology

Address for correspondence:

Prof. Igor M. Verner

Department of Education in Technology and Science

Technion - Israel Institute of TechnologyHaifa 32000, ISRAEL

Phone: 972-48292168

Email: [email protected]

Abstract

This paper presents two case studies that examine an approach to teaching geometry

through an ethnomathematics exercise in analysis and construction of culturally meaningful

ornaments. The exercise was given to students from the Arab sector high schools in Israel.

The studies indicated that the students perceived the practice of constructing geometrical

ornaments and discovery of their mathematical properties as a meaningful and enjoyable

learning experience. This experience inspired emotions, lively discourse, and learning

motivation. It arose into a geometrical and socio-cultural inquiry, reflecting the students' thirst

for practical use of the acquired mathematical knowledge and their awareness of cultural

identity.

Introduction

Ethnomathematics integrates mathematics and mathematical modeling with cultural

anthropology (Orey & Rosa, 2007). In this approach, problems from the learners' culture and

other cultures facilitate acquisition of mathematical knowledge and expose the learners to

commonalities across cultures and societies. As opposed to a value-free, culture-free

approach, ethnomathematics integrates mathematical practices historically developed in

different cultures and proposes a multicultural approach to education (Presmeg, 1998).

Multicultural education examines and implements approaches to create equal educational


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opportunities for students from diverse cultural groups, and impart abilities to communicate in

a pluralistic society and function for the common good (Banks and Banks, 1997, pp. 3-4).

The connection between ethnicity and mathematics manifests itself strongly in the

geometry of visual arts across cultures. Geometry is in the heart of every culture and is

inherent in every human mind. One may wonder from the many statements of recognition and

appreciation of geometry expressed by great scientists, artists, and philosophers throughout

history and across cultures. Since ancient times, geometric reasoning has been associated with

intelligence and truth:

"Who wishes correctly to learn the ways to measure surfaces and to divide them,

must necessarily thoroughly understand the general theorems of geometry andarithmetic, on which the teaching of measurement ... rests. If he has completely

mastered these ideas, he ... can never deviate from the truth."Abraham bar Hiyya (11th century Jewish mathematician,

astronomer and philosopher), in Treatise on Mensuration

"Geometry enlightens the intellect and sets one's mind right. All its proofs are very

clear and orderly. It is hardly possible for errors to enter into geometricalreasoning, because it is well arranged and orderly. Thus, the mind that constantly

applies itself to geometry is not likely to fall into error. In this convenient way, theperson who knows geometry acquires intelligence."

Ibn Khaldun (14th century Arab historian)

The universality of geometry in perceiving nature and expressing human feelings is greatly

acknowledged:

"The great book of Nature lies ever open before our eyes... But we cannot read itunless we have first learned the language and the characters in which it is written...It is written in mathematical language and the characters are triangles, circles and

other geometric figures...Galileo

"I have come to know that Geometry is at the very heart of feeling, and that each

expression of feeling is made by a movement governed by Geometry. Geometry is

everywhere in Nature. This is the Concert of Nature.Auguste Rodin


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Educators believe that incorporating applications of mathematical reasoning, inquiry and

modeling increases learners' motivation and creativity (Gravemeijer & Doorman, 1999) and

that this approach deserves educational research (Schoenfeld, 1998). Recent research supports

the view that modern education that emphasizes systems approach, project oriented learning,

cross-disciplinary linkages and multi-cultural contexts, can be enriched by proper engagement

of affect in the learning process (Picard et al., 2004, Goldin, 2006). This view is in line with

seminal observations that mathematical education cannot be reduced to "culture free"

mathematical training, rather it is "a process of inducting the young into part of their culture"

(Bishop, 1988).

This article presents a pilot experiment in which Arab school students in Israel studied

geometry through an ethnomathematics exercise of analysis and construction of geometrical

ornaments from their own and other cultures. We discuss learning processes through

geometric construction practices and cultural inquires of ornaments and summarize findings

of qualitative observations in two case studies.

The first case

Course description

In this case we designed, implemented and evaluated a 'Geometry with Applications'

course. The course was given by the first author to a 10th grade honors class (N=15) at a

school in an Arab village. The pre-course questionnaire indicated that most of the students did

not see the relation of mathematics to the real world. In order to expose the students to this

relation, while preserving the content and level of the geometry curriculum, the course

included a formal class and an optional supplementary workshop. In our case, all the students

opted for the workshop. In the class, different applied problems were used to illustrate the

studied concepts, while in the workshop the students practiced geometry through experiential

activities.


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In the first stage of the workshop, we offered applied geometry problems from different

areas of everyday life, science, and technology. The problems were selected as grounded in

the recommendations of the realistic mathematics education approach (Alsina, 1998;

Gravemeijer & Doorman, 1999). Students' feedback indicated that problems related to

culturally meaningful geometrical patterns were most motivating. The second stage of the

workshop focused on practical construction of geometrical ornaments by means of compass

and straightedge and the analysis of the construction steps. This analysis included identifying

basic geometrical objects, studying their properties and writing formal proofs. Figure 1A

presents an Islamic ornament studied in the workshop (Broug, 2009). The students

constructed the basic unit by means of compass and straightedge (Figure 1B), scanned it into

MS Word, and assembled the entire ornament by copy-paste operations.


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Figure 1: A. Sample ornament; B. Square unit

Construction and analysis of the ornament

We developed geometrical analysis activities in connection with the ornament construction

steps proposed by Broug (2009):

Step 1.

Construction. The students drew a circle and its vertical and horizontal diameters. Then they

drew a circumscribed square around the circle and its diagonals.

Constructing a perpendicular at a given point on the line segment was the

basic operation performed by compass and straightedge.

Analysis. The inquiry was guided by the following questions: (A) Why the circumscribed

quadrilateral is a square? (B) What are the symmetry axes of the basic unit? (C) What are the

values of the angles at the center of the circle?

Step 2.

Construction. The students drew four pairs of straight segments. Each of the segments starts

at the middle point of a side of the square, passes through the intersection

of the square's diagonal with the circle, and extends to the neighboring

side of the square.

A. B


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Analysis. The following questions were asked: (A) Is this octagon regular? (B) What are

values of its interior angles? How to calculate the values of interior angles of regular polygons

in general?

Step 3.

Construction. The students drew two quadrilaterals inscribed in the circle.

Analysis. Questions asked: (A) Are these quadrilaterals squares? (B) How

do you prove your claim using properties of the diagonals of an

quadrilateral? (C) Are the two squares congruent?

Step 4.

Construction. The students drew four segments. Each of the segments passes through two

points of the intersection of the squares constructed at Step 3, and its

endpoints lie on the sides of the circumscribed square constructed at Step 1.

Analysis. Inquiry task 1: Prove that at this step two pairs of parallel

segments are constructed.

Step 5.

Construction. The students drew two other pairs of parallel segments, horizontal and vertical.

The segments pass through the same intersection points, as at Step 4.

Analysis. Question asked: Explain why the segments in each of the pairs are

parallel. Why the quadrilateral formed by the segments is a square?

Step 6.

Construction. The students strengthen the straight segments constructed at Step 2 except for

the blank sections cut by the parallel segments.

Analysis. Inquiry task 2: Calculate the length of the blank section,

supposing that the radius of the circle is R.


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Geometrical inquiry tasks

Here we present solutions of the inquiry tasks 1 and 2 assigned in connection with Steps 4

and 6 of the ornament construction.

Solution of Task 1.

ABCDCB = (inscribed angles subtend equal arcs)

CEB is an isosceles triangle EBEC= .

CEF BGE because 90== EBGFCE ,

EBEC= , and GEBFEC = (vertical angles).

Similarly, we get:

...======= AHFACFECBEGBDG ...=== FHEFGE .

In the HCG : ,CECF = EGFH = FE HG . Similarly, MN KL . From MN FE, it

follows that HGKL .

Solution of Task 2.

Let Rbe the radius of the circle. Then, the length of

the chord ACsubtending the 45 arc equals

22 = RAC . This can be calculated using

trigonometry or geometrically from the similarity of

PCQ with AOC .

Denote xFACFECBE ==== . Then, 2= xEF because ECF is an isosceles right

triangle. The length of the chord AB can be calculated in two ways:

222 +== xxRAB , from which we conclude that12 +

=

Rx

= R( 2 1) .

A

B

C

D

E

F

K

M

G

H

LN

A

B

C

E

F

P

Q

O

S T

Y


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The isosceles triangles STF and ACO are similar, because ACOSTF = and

CAOTSF = (corresponding angles), therefore the sides are proportionalOC

FT

AC

ST= . By

substituting OCAC, we get:

From the similarity of the right triangles AYC and AFT , it followsYC

FT

AY

AF= . After

substituting to the proportion expressions of YCAYAF ,, by x :

2

2

2

2

=

+x

FT

xx

x, we

obtain

( )( )12222

++

=R

FT .

By substituting (2) in (1), we find the sought value of the blank section

RRST = 13.024158 .

Observation of learning activities

From the geometrical perspective, the course involved the students in a wide spectrum of

geometrical reasoning activities. While drawing segments and arcs of circles during the

construction process, the learners identified points of intersections, distances, perpendiculars,

tangent lines etc. The learners found symmetric components and designed repeated operations

for their construction. With the progress of the pattern construction, the learners perceived

geometrical objects formed as combinations of elements, such as pairs of lines, angles,

triangles, quadrilaterals, and polygons. They asked themselves questions that aimed to

characterize the objects:

- Are the lines parallel?

- Is this a right angle?

(1)

(2)


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- Is the triangle isosceles?

- Is the quadrilateral a square?

- Is the polygon regular?

They also compared pairs of geometrical objects with respect to characteristics such as

congruence, similarity and proportion.

From the cultural education perspective, as a first reaction, the students were surprised that

ornaments have a connection to geometry and are introduced in the geometry class. The

course motivated the students to search the web for additional ornaments, rooted in their own

culture, religion and environment. For example, the students found on the Web and brought to

class pictures of a mosque gate and grille decorated with geometrical ornaments. The students

recalled that these ornaments were the same as that decorating the mosque in their own

village. During the course they became conscious that the construction of ornaments is

grounded on universal geometrical concepts.

All the students actively participated in the workshop, consistently attending it and even

asking for extra activities. The follow-up indicated that practice with ornaments increased

students' interest and motivation to study geometry. Some of the students who were usually

passive in class actively participated in the workshop. The students were emotionally affected

by activities with ornaments connected with their own history and culture.

The second case

In this case study we developed and evaluated a pilot curriculum "Plane Geometry through

Ornament Analysis and Construction". The curriculum was approved by the superintendent of

mathematics education and implemented in an urban Arab sector school. The participants

were 10th grade students (N=35) studying intermediate level mathematics. The curriculum

was taught by the class teacher guided and followed up by the researchers. The case study


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aimed to answer the following research question: does the course affect changes in learners

attitudes, perceptions, and beliefs of geometry and cultural identity?

Curriculum outlineThe 13-hour curriculum consisted of geometry classes and extracurricular workshop

meetings. The topics and activities are presented in Table 1 and described below.

Table 1. Learning topics and activities

Topic/Activity Workshop ClassOrnaments in architecture and art historical and cultural

views. Interactive introduction to the geometry of an

ornament.

1-2

Basic geometrical figures: triangles, quadrilaterals, regular

polygons, and circles. Properties: congruence, rotation,

translation, reflection and symmetries.

3

Construction by straightedge and compass. 4-5Construction and analysis of an Islamic ornament by

straightedge and compass guided by the teacher. Home task.6-7

Further exercise and home assignment construction of an

advanced geometrical pattern.8

Presentation of the home assignment. 9Introduction to Rangoli (Hindu ornaments, their origins and

symbols). Teamwork on analysis and construction of a

Rangoli, driven by an instructional unit. Home assignment.

10-11

Presentation of team assignments. 12-13


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The first two class hours were taught by the researchers. Ornaments were introduced as

artifacts of symbolic meaning in art and culture created through geometrical construction. The

students performed an exercise in which they identified a basic unit of a typical Islamic

ornament and analyzed the unit's symmetries. They were asked to draw additional lines while

keeping the symmetries. In the second exercise, the students analyzed a photo of the ornament

that decorates one of the historical buildings in Egypt. They worked in groups and identified

the basic unit, its geometrical components, symmetries, and the way it is used to generate the

whole ornament.

During the third hour, the teacher reviewed the properties of basic geometrical figures

related to the construction of the Islamic ornament. For the next two hours, the teacher was

equipped with a compass and straightedge suitable for blackboard drawing. She introduced

the basic unit of the ornament to be constructed and explained that historically compass and

straightedge were the tools used for its construction. Then she involved the students in

analyzing the method of constructing the basic unit by compass and straightedge only.

The basic unit of the ornament was constructed during the workshop (hours 6-7). The next

workshop hour (8) and the following home assignment were devoted to an exercise in which

the students individually analyzed and constructed by compass and straightedge an advanced

geometrical pattern. The students presented their work during the next class meeting (hour 9).

At the end of the class the teacher introduced Rangoli ornaments of the Hindu traditional

culture - and the students got an instructional unit on Rangoli for the first look.

At the workshop (hours 10-11), the students worked in groups on constructing different

Rangolis by means of compass and straightedge. They discussed and asked guidance on the

following issues: symmetries, coloring, symbolic patterns in Rangolis, and their use and

interpretations in other cultures.


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selected to provide the construction of a certain ornament that she had in mind. Next, the

teacher introduced the ornament. The students were first asked to observe the basic unit and

analyze the ornament, that is, to detect the way in which it was constructed using the above

mentioned construction operations. The students, by themselves, recognized more than one

method to do so. After that, they worked in groups on the construction of the basic unit

(Figure 2) by means of compass and straightedge. The groups challenged each other.

Figure 2. Islamic ornament basic unit constructed by the students

The students recognized the need to perform constructional operations of drawing a

perpendicular to a straight line, bisecting a given angle and bisecting a segment. They were

taught the operations needed for constructing the ornament. The construction operations were

substantiated by geometrical proofs.

The final stage was devoted to activities with Rangolis. The students worked in groups and

performed the tasks given in the instructional unit prepared by the authors. They constructed

Rangolis by compass and straightedge and colored them using traditional combinations of

colors that emphasize the ornaments' symmetries.

An example of Rangoli constructed on a square dot grid and colored by one of the groups

is presented in Figure 3.


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A B

Figure 3. Rangoli: A. Construction of pattern; B. Colored pattern

The problems solved in connection to this Rangoli are presented below.

Problem 1. Let the square dot unit in Figure 3.A be . Express the length of the

segments and in terms of .

Solution: is perpendicular to and bisects it .

(radii of the circle ). ( is right angled).

(as ).

So we get:

;

.

Below in Problem 2, we will use the notations1 , , 4 , , , to denote the

geometric domains that encompass these symbols in Fig. 3B.

Problem 2. Let and indicate reflections over the X and Y axes, indicates the

C

P

K1

O

I

KE FD

C

H

G


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reflection over the line xy = , S indicates the counterclockwise quarter-turn about the point

O, and T indicates a half-turn about the point O.Answer the following questions:

Q1. What are the images of )(),(),(),( 43 PRPSKSPSX

?Q2. What are the images of

Q3. Suggest a series (composition) of transformations that transfers2

to4

,2

to1 .

Q4. Answer true or false for the following claims:

62 )( =T ,

45 )( =K , =K any combination ofT 's andR 's.

Sample answers:

In Q1, PKS =)(3 , meaning that the operation onKis three consecutive counter clock-wise

quarter-turns, which movesKtoP.

In Q2, 411 )( =SSR

X. The operation is a sequence of: (1) a counter clock-wise quarter-turn

which moves1 to

4 , (2) reflection over the X axis which moves 4 to 1 (note that

XX

RR =1

).

In Q3, a possible composition of transformations from2

to4

is

422 ))(()( = == XYXXYX RRRR .

A possible one-transformation solution is 42 )( =S .

Pre-Instruction Results

Before running the curriculum, a survey examining students' reflection on their experience

of geometry studies was conducted among 10th grade students in two classes from different

Arab sector schools. One was an advanced level mathematics class from a selective school

(N=25) taught by the first author. Another was an intermediate level mathematics class from a

heterogeneous school (N=28). The students were unaware of our intention to run the

curriculum. The latter class was selected for the experiment.


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The survey referred to students' interest in cultural applications of geometry and examined

if the students recognized geometrical objects when seeing artifacts of architecture, design,

and art. The survey findings indicated what follows:

All students in the advanced mathematics class and the majority of students in theintermediate mathematics class (79%) consider geometry to be an important subject for

them. The reason pointed out by the absolute majority of the first group (88%) and the

second group (96%) is that it is a compulsory matriculation subject.

The majority of the students in the first group (72%) and only35% of the second groupconsider geometry to be an interesting subject.

Some students in the first group (16%) and the majority of the second group (57%) donot recognize geometry to be instrumental in their future studies.

Some students of the first group (20%) and almost half of the second group (43%)believe that geometry does not contribute to personal success in life.

Some students of the first group (16%) and a significant part of the second group (32%)do not see the connection between geometry and graphical arts (architecture, graphics,

painting, etc.).

Very few students of the first group (8%) and half of the second group (50%)complained that geometry is taught as a theoretical and abstract subject.

The survey results were discussed with the intermediate class teacher. She was aware about

the need of change in teaching geometry, got interested in our curriculum and decided to

implement it in her class.


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Post-Instruction Findings

The follow-up included observations of learning activities, written reflections, talks and

interviews with students and the teacher during and after the experiment. Here is a summary

of findings from student reflections:

All the students accurately attended the course meetings. They even asked to extend thecourse.

The students, including those who usually were inactive in regular geometry class,expressed enjoyment of the learning activities in class and workshop alike.

The students drew ornaments and analyzed every step formally, using concepts andtheorems studied in the regular geometry class.

The students, on their own, searched the web for ornaments. They brought to classexamples related to their religion and national culture. Some of the students were

enthusiastic to show the class ornaments that they personally faced in the past.

In the geometry class during and after the experiment the students were curious to seehow the new concepts and theorems could be interpreted in the context of ornaments.

Ornaments became a subject of the students' discourse also after classes and at leisuretime.

Typical quotations from the students' interviews:

Salam: First time ever that I understand geometry.

Yusof: I discovered that geometry has a special magic and that it is important.

Nimr: I very much enjoyed it. The group work drew us close.

Hanna: Not only theorems and proofs it is an enjoyable experience of discovering

and drawing.

Ranya: I would prefer to study geometry this way.


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Hanin: In the past I did not pay attention to ornaments. Now I examine how they are

constructed.

Yusof: Ornaments are beautiful artistic designs. The mind uses straightedge and

compass artistically.

Mahmud: It opened my eyes towards other cultures that were unknown to me.

Also, here are some observations about the teachers attitude:

Enthusiastic and active collaboration in implementing the curriculum. Interest in extra-curricular activities, such as drawing with compass and straightedge

and searching databases for ornaments of different cultures.

Perceiving the value of ornament analysis and construction for teaching geometry. The teacher said: I appreciate the opportunity to involve my students in this

informal activity. I myself learned new things. You opened my eyes.

Discussion and Conclusion

The experience of the pilot courses reveals a wide spectrum of geometrical reasoning

activities related to the analysis and construction of ornaments. While drawing segments and

arcs of circles during the construction process, the learners identify points of intersections,

distances, perpendiculars, tangent lines etc. Moreover, the learners find symmetric

components and design repeated operations for their construction. With the progress of

constructing the pattern, the learners perceive geometrical objects formed as combinations of

elements, such as pairs of parallel lines, triangles, quadrilaterals, and polygons.

They ask themselves questions about object's characteristics:

- Are the lines parallel?

- Is the triangle isosceles or right?

- Is the quadrilateral a square?

- Is the polygon regular?


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They also compare geometrical objects by examining such characteristics as congruence,

similarity and proportion.

Our educational experiments indicate that students and teachers from the Arab sector

schools in Israel perceived the learning and teaching practice with geometrical ornaments as

"an enjoyable experience of discovering and drawing". Due to this experience with real

ornaments, the students began to recognize them as geometrical objects. They became

interested in observing real structures and capable of exploring their geometrical properties.

The distance between "school geometry" and "real world" in students' beliefs is reduced.

Moreover, practice with ornaments aroused students' awareness and motivated inquiry of

their historical roots and cultural value. The students searched for answers to:

- Where we can see the ornament in reality?

- Does it belong to my culture? If yes, how do I recognize that? If not, to what culture it

belongs?

- What does it symbolize?

- Can we face similar ornaments in other cultures? If yes, do they have the same symbolic

meaning?

- What shapes and colors are traditional in my culture?

To our surprise, teaching geometry through ornaments, as opposed to other applications,

inspired the students and teachers with a flow of emotions, lively discourse, and learning

motivation. A detailed analysis of this behavior cuts across psychology, anthropology and

education, and it is beyond our study. However, we observed that activities with ornaments

arose into a socio-cultural inquiry, reflecting spiritual needs and awareness of personal and

cultural identity. In any case, our findings are in line with the view of multiculturalism

researchers (Moghaddam, 2008) and mathematics educators (Saxe, 1991) that socio-cultural


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inquiry associated with artifacts' construction can facilitate learning and cognitive

development.

Acknowledgement

This research is partially supported by the Ministry of Science and Technology, Israel. The

authors thank Daniel Orey for helpful constructive comments.

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Bishop, A. J. (1988). Mathematics education in its cultural context. Educational Studies in

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an ethnomathematics program.For the Learning of Mathematics, 27(1), 10-16.Picard, R. W., Papert, S., Bender, W., Blumberg, B., Breazeal, C., Cavallo, D., Machover, T.,

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Documents

An Ethnomathematics Exercise in Analyzing and Constructing Ornaments